SELF

84

S.B. Karavashkin, O.N. Karavashkina

The limiting process to a distributed line can be effected on the basis of yielded solutions under the condition 0, if the main parameters characterising the line were presented in the form convenient for it, i.e.

where  x₀ is the distance from the reading-point to the point of rest of the considered line element.

Besides, the necessary requirement is, the values , T and x₀   to remain finite in the limiting process. For example, for the solution (50) we yield

where x_0i  is the location of studied point at rest, x_0k  is the location of external force application point, also at rest, x_i is the momentary location of investigated point in the excited state; l  is the line length, and _i is the line element shift.

As follows from (73), when transiting to a distributed line, there remains the main distinction caused by the external force application point that divides the line into sections. Consequently, on the whole the vibration structure remains and, at definite conditions, the vibrations are produced only in a part of a line. With it, when passing to the limit, (50) transforms irreversibly, which makes the reverse transition impossible.

When studying the conditions of transition to a distributed line, it is interesting to consider, how the solutions for critical and aperiodical regimes transform. Conveniently consider this not on the basis of particular solutions but under the condition of these regimes realisation. When a 0 , noting (53) and (72), we yield

whence it follows that of all three vibration regimes only that periodical can be realised with the limiting process. For two other regimes the conditions of existence will be simply absent.

Thus, analysing the transition to a distributed line, we see disappearing the number of properties inherent namely in lumped lines, which makes descriptions of these models non-identical. We can pass from the lumped line description to that distributed, but vice versa we cannot. Only some particular approximation, when the condition (71) was fulfilled, is admissible.

Conclusions

In the course of study we have revealed that not only the line finiteness, the way of the ends fixation or vibration regime have a great effect on the vibration pattern but also the external force application point. In particular, this point is able to be a border of a quasi-finite section where a standing wave forms, to form the vibrations of complicated structure and conditions at which the vibration process fully vanishes in one of sections of the line, while other sections retain the vibrations. This all gives the grounds to indicate the feature of heterogeneity, which the external force application point causes in the line.

We also have revealed that in lumped lines, with the growing vibration frequency, the progressive wave propagation velocity decreases nonlinearly and monotonously, reaches the minimal value at _crit  and retains it in the aperiodical regime.

Similarly, the wavelength in the line decreases nonlinearly to the double distance between the line elements, and it retains this value in the aperiodical regime.

At the limit passing to the distributed line, the aperiodical and critical vibration regimes become impossible. Besides, at this passing, the main distinctions of solutions that are caused by the external force affection on the interior elements retain. At the same time, when transiting to a distributed line, the solutions transform irreversibly, which makes the reverse transition impossible.

We have established also that a lumped line can be modelled by means of that distributed only when the ratio of between-the-elements distance to the wavelength is considerably less than ^-1 .